In order to obtain a robust characterisation of stellar motions, we need information of proper motions and distances. The proper motions are obtained from the Gaia EDR3 ([1]}; [2]}), which provides the best astrometric data nowadays. We adopt the criteria \(\rm {ASTROMETRIC_{-}GOF_{-}AL}<3.0\)  and \(\rm {ASTROMETRIC_{-}EXCESS_{-}NOISE_{-}SIG}\leqslant 2.0\)  to select astrometrically well-behaved stars with good-fit statistic of the astrometric solution, coupled with a selection of the Renormalised Unit Weight Error (RUWE) \(<1.4\)  [3]} to favour stars with good astrometric quality. The distances, for the geometry, can be simply estimated by the inverse of the trigonometric parallax. However, such a method is feasible only for parallaxes with very high precision ([4]}; [5]}). Therefore, we employ the photometric distances derived from the measurements of distance modulus. To do this, we adopt the absolute magnitudes estimated using the method of [6]}, which is tailored specifically to the spectroscopically derived stellar parameters of LAMOST. Comparison with the apparent magnitude, which is obtained from the 2MASS point source catalog ([7]}; [8]}) in the \(K_s\)  band (2.16 \(\mu \) m), yields the distance modulus and then the distance. In order to correct for the extinction, we adopt \(R(K_s)=0.306\)  from [9]} as the reddening relative to \(E(B-V)\)  obtained from [10]}. We obtain 545227 K giants with measurements of proper motions and distances.
